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<title>Normal Forms +&nbsp; Simple Pendulum Example</title>
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<p>Type NORMALFORM&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (Factorization into a canonical 
form of the map N%A_T <a href="factor_a.htm">click here</a>)</p>

<p>This type is central to accelerator physics. Here I reproduce in jpeg form 
some transparenci&nbsp;&nbsp; es from a PowerPoint presentation.</p>
<p>This presentation culminates with a program computing the dependence of a 
pendulum frequency with amplitude. This is obtained with the type NORMALFORM. 
The program is available <b><a href="../simple_programs/z_pendulum.f90">here</a></b>.</p>
<p>A Twiss function &quot;do-loop&quot; is displayed in the following PTC example: you 
must download <a href="als.txt">a flat file of the ALS light source</a> and
<a href="z_als_example_twiss.f90">a main program</a>. Link it with FPP and PTC.</p>
<ol>
  <li><font size="5">The Normal form Type</font></li>
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    <td width="50%"><a href="n1.jpg">
    <img border="0" src="n1.jpg" width="375" height="277"></a></td>
    <td width="50%"><a href="n2.jpg">
    <img border="0" src="n2.jpg" width="375" height="279"></a></td>
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    <td width="50%"><a href="n3.jpg">
    <img border="0" src="n3.jpg" width="375" height="279"></a></td>
    <td width="50%"><a href="n4.jpg">
    <img border="0" src="n4.jpg" width="375" height="279"></a></td>
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&nbsp;<p>&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
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  <li><font size="5">The example of a simple pendulum</font></li>
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    <td width="50%"><a href="pendulum1.jpg">
    <img border="0" src="pendulum1.jpg" width="400" height="299"></a></td>
    <td width="50%"><a href="pendulum2.jpg">
    <img border="0" src="pendulum2.jpg" width="400" height="299"></a></td>
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    <td width="50%"><a href="pendulum3.jpg">
    <img border="0" src="pendulum3.jpg" width="400" height="299"></a></td>
    <td width="50%">&nbsp;</td>
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